License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2022.71
URN: urn:nbn:de:0030-drops-164123
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16412/
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Harms, Nathaniel ; Yoshida, Yuichi

Downsampling for Testing and Learning in Product Distributions

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LIPIcs-ICALP-2022-71.pdf (0.8 MB)

Abstract

We study distribution-free property testing and learning problems where the unknown probability distribution is a product distribution over ℝ^d. For many important classes of functions, such as intersections of halfspaces, polynomial threshold functions, convex sets, and k-alternating functions, the known algorithms either have complexity that depends on the support size of the distribution, or are proven to work only for specific examples of product distributions. We introduce a general method, which we call downsampling, that resolves these issues. Downsampling uses a notion of "rectilinear isoperimetry" for product distributions, which further strengthens the connection between isoperimetry, testing and learning. Using this technique, we attain new efficient distribution-free algorithms under product distributions on ℝ^d:
1) A simpler proof for non-adaptive, one-sided monotonicity testing of functions [n]^d → {0,1}, and improved sample complexity for testing monotonicity over unknown product distributions, from O(d⁷) [Black, Chakrabarty, & Seshadhri, SODA 2020] to O(d³).
2) Polynomial-time agnostic learning algorithms for functions of a constant number of halfspaces, and constant-degree polynomial threshold functions;
3) An exp{O(dlog(dk))}-time agnostic learning algorithm, and an exp{O(dlog(dk))}-sample tolerant tester, for functions of k convex sets; and a 2^O(d) sample-based one-sided tester for convex sets;
4) An exp{O(k√d)}-time agnostic learning algorithm for k-alternating functions, and a sample-based tolerant tester with the same complexity.

BibTeX - Entry

@InProceedings{harms_et_al:LIPIcs.ICALP.2022.71,
  author =	{Harms, Nathaniel and Yoshida, Yuichi},
  title =	{{Downsampling for Testing and Learning in Product Distributions}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{71:1--71:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16412},
  URN =		{urn:nbn:de:0030-drops-164123},
  doi =		{10.4230/LIPIcs.ICALP.2022.71},
  annote =	{Keywords: property testing, learning, monotonicity, halfspaces, intersections of halfspaces, polynomial threshold functions}
}

Keywords: property testing, learning, monotonicity, halfspaces, intersections of halfspaces, polynomial threshold functions
Collection: 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)
Issue Date: 2022
Date of publication: 28.06.2022

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